Optimization theory • Nonlinear least squares __ optimization

Source: Internet
Author: User
Optimization theory • Nonlinear least squares

tags (space-delimited): Math

The nonlinear least squares problem is one of the most easily encountered optimization problems in elliptic fitting, this paper mainly introduces the basic analysis of nonlinear two-squares 1. What is the least squares problem

The objective function can be written as the optimization problem of the square sum of M functions

where each function fi (x) f_i (x) is a function to optimize vector x x. 2. Nonlinear least squares problem when fi (x) f_i (x) is a nonlinear function of x x, it is a nonlinear optimization problem at this point, we need to use Taylor first-order expansion approximation fi (x) f_i (x) 2.1 fi (x) f_i (x) First-order approximation

Set X (k) x^{(k)} to be the K-th approximation of the solution x x, where fi (x) f_i (x) is first expanded and the first-order expansion value is φi (x) \varphi_i (x)

The upper type is sorted and

As you can see, the first-order approximate φi (x) \varphi_i (x) is a linear function of the vector x x to be optimized: Is fi (x) f_i (x) Gradient "fi (x) f_i (x) to derivation of vector X" at x (k) x^{(k)} The value is fi (x) f_i (x) in X ( k) approximation of the value 2.2 F (x) at x^{(k)}

After obtaining the first order approximation of fi (x) f_i (x), we can calculate the first order approximation of f (x), which is approximately Φ (x

Contact Us

The content source of this page is from Internet, which doesn't represent Alibaba Cloud's opinion; products and services mentioned on that page don't have any relationship with Alibaba Cloud. If the content of the page makes you feel confusing, please write us an email, we will handle the problem within 5 days after receiving your email.

If you find any instances of plagiarism from the community, please send an email to: info-contact@alibabacloud.com and provide relevant evidence. A staff member will contact you within 5 working days.

A Free Trial That Lets You Build Big!

Start building with 50+ products and up to 12 months usage for Elastic Compute Service

  • Sales Support

    1 on 1 presale consultation

  • After-Sales Support

    24/7 Technical Support 6 Free Tickets per Quarter Faster Response

  • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.